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The Hadwiger number of a graph $G$, denoted $h(G)$, is the largest integer$t$ such that $G$ contains $K_t$ as a minor. A famous conjecture due toHadwiger in 1943 states that for every graph $G$, $h(G) \ge \chi(G)$, where$\chi(G)$ denotes the chromatic number of $G$. Let $\alpha(G)$ denote theindependence number of $G$. A graph is $H$-free if it does not contain thegraph $H$ as an induced subgraph. In 2003, Plummer, Stiebitz and Toft provedthat $h(G) \ge \chi(G)$ for all $H$-free graphs $G$ with $\alpha(G) \le 2$,where $H$ is any graph on four vertices with $\alpha(H) \le 2$, $H=C_5$, or $H$is a particular graph on seven vertices. In 2010, Kriesell considered aparticular strengthening of Hadwiger's conjecture due to Seymour andsubsequently generalized the statement to include all forbidden subgraphs $H$on five vertices with $\alpha(H) \le 2$. In this note, we prove that $h(G) \ge\chi(G)$ for all $W_5$-free graphs $G$ with $\alpha(G) \le 2$, where $W_5$denotes the wheel on six vertices.

A note on Hadwiger's Conjecture for $W_5$-free graphs with independence number two